Nuprl Lemma : qrng_wf
<ℚ+*> ∈ CRng
Proof
Definitions occuring in Statement :
qrng: <ℚ+*>,
member: t ∈ T,
crng: CRng
Definitions unfolded in proof :
member: t ∈ T,
crng: CRng,
comm: Comm(T;op),
qrng: <ℚ+*>,
rng_car: |r|,
pi1: fst(t),
rng_times: *,
pi2: snd(t),
infix_ap: x f y,
uall: ∀[x:A]. B[x],
rng: Rng,
prop: ℙ,
rng_sig: RngSig,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_plus: +r,
rng_zero: 0,
rng_minus: -r,
rng_one: 1,
cand: A c∧ B,
group_p: IsGroup(T;op;id;inv),
rationals: ℚ,
quotient: x,y:A//B[x; y],
grp_car: |g|,
qadd_grp: <ℚ+>,
grp_op: *,
grp_id: e,
grp_inv: ~,
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
squash: ↓T,
true: True,
bilinear: BiLinear(T;pl;tm)
Lemmas referenced :
qmul_com,
rationals_wf,
comm_wf,
rng_car_wf,
rng_times_wf,
ring_p_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_one_wf,
qeq_wf2,
q_le_wf,
qadd_wf,
int-subtype-rationals,
qmul_wf,
eqtt_to_assert,
assert-qeq,
it_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
qdiv_wf,
unit_wf2,
imon_properties,
qadd_grp_wf2,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
igrp_properties,
grp_subtype_igrp,
abgrp_subtype_grp,
ocgrp_subtype_abgrp,
abgrp_wf,
grp_wf,
igrp_wf,
qmul_assoc,
equal_wf,
iff_weakening_equal,
qmul_ident,
q_distrib
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality_alt,
sqequalRule,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
setElimination,
rename,
because_Cache,
dependent_pairEquality_alt,
lambdaEquality_alt,
closedConclusion,
natural_numberEquality,
applyEquality,
minusEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
inrEquality_alt,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
baseClosed,
inlEquality_alt,
functionIsType,
unionIsType,
productIsType,
independent_pairFormation,
imageElimination,
imageMemberEquality,
independent_pairEquality
Latex:
<\mBbbQ{}+*> \mmember{} CRng
Date html generated:
2020_05_20-AM-09_15_19
Last ObjectModification:
2020_01_18-AM-10_21_54
Theory : rationals
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